Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
2
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1answer
370 views

Splitting field that isn't a Galois extension

I'm trying to find a counter-example to following statement: if $K$ is the splitting field of $g\in F[x],$ then the extension $K/F$ is Galois. I know the statement is true if $g$ is separable, ...
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4answers
567 views

Minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$

Is there an "easy" way to find the minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$ without the help of any computer programme? If I knew $\sin(\pi/8)=\frac{\sqrt{2-\sqrt2}}{2}$ then it would ...
3
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1answer
107 views

Galois group of a quintic

What is the Galois group of $x^5-4x+12$? I'm able to show it has to be either the Frobenius $F_{20}$ group, or the dihedral group $D_{10}$. Is there a less computationally heavy way to determine? What ...
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1answer
74 views

galois theory radical extension..

If p(x) is solvable by radicals over F; then prove that there is a radical tower F = B0 B1    Bt such that Bt is a splitting field of some polynomial over F (or Bt=F is a normal extension).
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1answer
111 views

Galois theory Radical extension

Prove that any splitting field $K/F$ containing a radical extension $R_{t}/F$ is itself a radical extension.
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535 views

$\mathbb Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbb Q(\sqrt 2+\sqrt 3+\sqrt 5)$ - Generalisation?

Problem: We know that $\mathbb Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbb Q(\sqrt 2+\sqrt 3+\sqrt 5)$. Generalise this fact. My idea is to use the (proof of the) primitive element theorem. Looking at ...
4
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1answer
54 views

No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$

Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
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1answer
359 views

Degree of the splitting field of $x^{p^2} -2$ over $\mathbb{Q}$, for prime p.

I've already shown that the degree of the splitting field of $x^p-2$ over $\mathbb{Q}$ is $p(p-1)$ as follows: $x^p-2$ has roots $\sqrt[p]{2}\omega_{k}$ for $k=0,1,...,p-1$, where the $\omega_{k}$ ...
5
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2answers
138 views

$f \in K[X]$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?

Let $K$ be a field and $f \in K[X]$ of degree $n$ with Galois group $S_n$. Let $a$ be a root of $f$, $L = K(a)$, and let $E$ be a intermediate field of the extension $K \subset L$. Prove that $E = K$ ...
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0answers
89 views

Galois group of $X^4+4X^3+12X^2+24X+24$

What is the best method to find the Galois group of $P = X^4+4X^3+12X^2+24X+24$ over $\mathbb{Q}$ ? First, I don't manage to show that $P$ is irreducible : Eisenstein doesn't work. I know that its ...
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1answer
136 views

Question about separable extension

Here is an assignment problem.$\\$ Let $K/F$ be a finite field extension and $S=\{u\in K\ |\ \sigma(u)=u\ ,\forall \sigma\in \operatorname{Gal}(K/F)\}$. Suppose $S=F$. Prove or disprove that $K/F$ ...
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0answers
211 views

Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
4
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1answer
149 views

How to prove that this infinite extension of $\mathbb{Q}$ is Galois

Let $K_0=\mathbb{Q}$ and for $n>0$ define $K_{n+1}$ as the extension of $K_n$ obtained by adjoining to $K_n$ all the radicals of elements in $K_n$. Let $K$ be the union of the subfields $K_n$. ...
1
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1answer
47 views

Degree of the splitting field

I'm looking for a proof that, if $K/k$ is the splitting field of a polynomial $P$, and $z_1, ..., z_n$ the roots of $P$ in an algebraic closure of $k$, then $$[K : k] | \prod_{i=1}^n [k[z_i]:k]$$ Is ...
2
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0answers
198 views

Galois Theory References

This may not initially be a well posed question, but I'm looking for a good reference on Galois theory that covers it from the viewpoint of the symmetry in roots of an irreducible polynomial and not a ...
6
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2answers
150 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
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0answers
119 views

Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)

With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where ...
3
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1answer
376 views

Proving $ x^p - a $ is irreducible iff $ a $ is not a $p$-th power of an element

I have the following homework question, which I've managed to do the forward implication of, but not the other direction: Let $\mathbb{K}$ be a field, $ a\in\mathbb{K}$ and $p$ be a prime. Show that ...
2
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4answers
93 views

Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.

Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$. What I've done so far: Let $\phi(n) = p$ so the following products each have p components. $$\Phi_{n}(x) = \prod_{k=1, ...
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1answer
286 views

A radical extension with a non-radical subextension

For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
2
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1answer
168 views

Show the extension is not cyclic

Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
2
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3answers
90 views

Two quadratic fields over $\mathbb{Q}$

I'm having a bit of trouble showing that the two quadratic fields $\mathbb{Q}[X]/(X^2+1)$ and $\mathbb{Q}[X]/(X^2+3)$ over $\mathbb{Q}$ are not isomorphic (as fields). Could someone help me? Perhaps ...
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2answers
80 views

Larger Theory for root formula

Consider the quadratic equation: $$ax^2 + bx + c = 0$$ and the linear equation: $$bx + c = 0$$. We note the solution of the linear equation is $$x = -\frac{c}{b}.$$ We note the solution of the ...
2
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1answer
225 views

Fixed field of automorphisms determined by $t\mapsto at+b$.

Suppose $E=\mathbb{F}_p(t)$, the field of rational functions in a transcendental $t$ over the finite field of $p$ elements. Suppose $G$ is the group of field automorphisms fixing $\mathbb{F}_p$ ...
6
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1answer
257 views

Irreducibility over $\mathbb F_p$ - A useless hint?

Dummit and Foote, 13.5.5: For any prime $p$ and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$ The question goes on to suggest two approaches ...
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1answer
855 views

Find the Galois group of $x^3 -2$ over $\mathbb{Q}$. [duplicate]

Show that it is a non-abelian group of order 6, then under the Galois correspondence find the (fixed) subfield corresponding to the subgroup of G of order 3. I've found the splitting field which is ...
6
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2answers
139 views

Factoring $x^{255} -1 $ over $\Bbb F_2$

How would I factor the above polynomial in this binary field? We just completed a course in Galois Theory, and I'm stuck on how to efficiently factor this polynomial. I tried considering computing all ...
5
votes
2answers
432 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
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1answer
117 views

Definition of Galois Group

I'm revising for a module on Galois Theory and have trouble understanding the definition for a Galois group of a field extension $K:F$. Define the Galois group of $K:F$ as $\Gamma(K:F)=\{\sigma \in ...
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1answer
217 views

Fixed field of group of automorphisms on $\mathbb{C}(t)$.

I've been stuck on this problem tonight. Suppose $E=\mathbb{C}(t)$ where $t$ is transcendental over $\mathbb{C}$, and let $\omega$ be a primitive cube root of unity. Let $\sigma$ be the automorphism ...
1
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1answer
207 views

Generator of rational functions unchanged under $\sigma(X) = X + 1$

Let $L = K(X)$ be the field of rational functions over a field $K$ with characteristic $p > 0$, and let $\sigma \in \operatorname{Aut}_K(L)$ with $\sigma(X) = X + 1$. Show that $G = ...
3
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2answers
644 views

How to find the splitting field and Galois group of $x^6 -4x^3 +1$?

I am trying to find the splitting field $L$ of the $x^6 -4x^3 +1$ over $\mathbb{Q}$, and its Galois group. Here are some things I have figured out. I did the usual trick of solving for $x^3 = 2\pm ...
3
votes
2answers
115 views

Question on relation between normal subgroups and normal extensions in Fundamental Theorem of Galois Theory.

I'm self studying Jacobson's Basic Algebra I but I'm getting hung up on the proof of the Fundamental Theorem of Galois Theory in Jacobson's book on page 239. Let $G=\operatorname{Gal}(E/F)$ for a ...
5
votes
1answer
107 views

Unsolvability of $S_{n}$

Is there a short proof for unsolvability of $S_{n}$ without the standard approach of proving the simplicity of $A_{n}$ ? This is good, however, and one can prove this only with basic group theory, ...
1
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1answer
57 views

Why is $\operatorname{Gal}(E/F)\cong GL_2(F)/F^\ast$ when $E=F(t)$ for $t$ transcendental?

This is an example on page $235$ of Jacobson's Algebra I that I'm reading. I quote Let $F$ be a field and $E=F(t)$ where $t$ is transcendental over $F$. $u\in E$ is a generator of $E/F$ if and ...
3
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1answer
122 views

Walsh spectrum of a function defined over Galois rings

Let $GR(p^2,m)$ be the Galois ring with $p^{2m}$ elements and characteristic $p^2$. Let $Z^m_{p^2}$ be the cross product of $m$ copies of $Z_{p^2}$ which is the set of integers from zero up to ...
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1answer
133 views

Can any monomorphism of a subfield of a splitting field be extended to an automorphism?

It's a common theorem in field theory that if $\varphi: F\to\overline{F}$ is a field isomorphism, and if $E$ and $\overline{E}$ are splitting fields of monic polynomials $f(x)$ and $\overline{f}(x)$, ...
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Galois automorphisms and Field extensions [duplicate]

Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. I want to show that $\alpha^2+\beta$ has degree $9$. There are many ways to do this, but I wish to solve the problem ...
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0answers
111 views

Finite/algebraic extensions of rational functions

I'm looking for results on the subject of finite/algebraic extensions of rational functions, but I only find papers who deal with algebraic geometry. I only know the basis of Galois theory. Could you ...
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1answer
563 views

field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
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1answer
98 views

Find the closure and then the galois group

So today on my test I had this problem: Find the normal closure and its Galois group of $\mathbb{Q}(\sqrt{-3}+\sqrt[3]{2})/\mathbb{Q}$. I managed to find the minimal polynomial during the test and ...
2
votes
1answer
162 views

Galois Theory Problem (Fundamental theorem of Galois)

Let $k$ be a field of characteristic$>2$. Let $c\in k$, $c\notin k^2$. Let $F=k(\sqrt{c})$. Let $\alpha=a+b\sqrt{c}$ with $a,b\in k$ and not both $a,b=0$. Ket $E=F(\sqrt{\alpha})$. Prove that the ...
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1answer
131 views

Prove: Let $Gal(f)$ acts transitively on $Z(f)$ if and only if $f$ is irreducible in $F[x]$

Can someone provide a proof for this, please? Particularly for the backward direction. Let $F$ be a field. Let $f(x)$ be a separable polynomial in $F[x]$. Let $K/F$ be the splitting field of $f(x)$. ...
5
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1answer
249 views

The Galois group for the following field extension

I am trying to find the Galois group of the extension $\mathbb{Q}(\sqrt{2},\sqrt{3},\alpha)=K$ over $\mathbb{Q}$ where $\alpha$ is such that $\alpha^2=(9-5\sqrt{3})(2-\sqrt{2})$. Here is my attempt: ...
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0answers
198 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
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3answers
810 views

Why does rational trigonometry not work over a field of Characteristic 2?

I am interested in solving triangles in a finite field with a computer program. Rational trigonometry seems well suited to do this. However, the Wikipedia article, as well as several published ...
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0answers
143 views

Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
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1answer
375 views

Galois group of a cyclotomic extension

Do you know a condition on the field $k$ for that the injection of $\text{Gal}(k[\zeta_n]/k)$ in $(\Bbb Z/n\Bbb Z)^*$ is bijective ? It is the case for $k = \mathbb{Q}$, but not for $k = \mathbb{R}$ ...
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1answer
224 views

Example, Field of rational functions and a special automorphism group

I am refering to an example from the famous book Galois Thery by Emil Artin. There on page 38 he gave the following example as an application of the Theorem. (Theorem 13) If $\sigma_1, \ldots, ...